N = 2 SUPERSYMMETRIC YANG–MILLS AND THE QUANTUM HALL EFFECT

Abstract

It is argued that there are strong similarities between the infrared physics of N = 2 supersymmetric Yang–Mills and that of the quantum Hall effect, both systems exhibit a hierarchy of vacua with a subgroup of the modular group mapping between them. The coupling flow for pure SU(2) N = 2 supersymmetric Yang–Mills in four dimensions is reexamined and an earlier suggestion in the literature, that was singular at strong coupling, is modified to a form that is well behaved at both weak and strong coupling and describes the crossover in an analytic fashion. Similarities between the phase diagram and the flow of SUSY Yang–Mills and that of the quantum Hall effect are then described, with the Hall conductivity in the latter playing the role of the θ-parameter in the former. Hall plateaux, with odd denominator filling fractions, are analogous to fixed points at strong coupling in N = 2 SUSY Yang–Mills, where the massless degrees of freedom carry an odd monopole charge.